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Wednesday, March 27, 2013

Probably since forever philosophers and mathematicians have
dreamed of mechanizing thought, of removing judgment from thinking. The newest
aspirant in this millennial quest is Big Data, and not surprising there is an
eponymous book (excerpted here) with the following provided summary:

This revelatory exploration of big
data, which refers to our newfound ability to crunch vast amounts of
information, analyze it instantly and draw profound and surprising conclusions
from it, discusses how it will change our lives and what we can do to protect
ourselves from its hazards.

Big Data (BD) is the new New Thing, the method by which diligence
can substitute for thought. The idea actually has a certain charm as it
reverberates with our sense of justice. Collecting data is hard work, but it is
generally the kind of work for which effort is rewarded. Work hard and you will
do well. Put in the hours and the data will pile up. It’s an activity that rewards virtue.

In this it is entirely unlike
coming up with a plausible analysis, aka thinking. This activity is totally
unfair. Lazy people can have excellent
ideas. Sloth is no bar to insight and profligacy no guarantee of intellectual
stagnation. Here even the wicked,
sloppy, and lazy can prosper. How unfair.

In a just world, virtue would be rewarded. In a just world
hard work would guarantee enlightenment. We don’t live in a just world. Big
Data is the unfounded belief that this can be remedied. The hard work of data
gathering can substitute for the caprice of thought. It cannot be, and,
unfortunately, believing it can is likely to deform scientific practice. To see this, consider the following quote:

The era of big data challenges the
way we live and interact with the world. Most strikingly, society will need to
shed some of its obsession for causality in exchange for simple correlations:
not knowing why but only what. This overturns centuries of
established practices and challenges our most basic understanding of how to
make decisions and comprehend reality (10).

And that is precisely the problem. Big Data is part of an enterprise aimed at
reforming scientific practice. Dump why
aim for what. However, contrary to the prevailing
conception, without a model/theory it is not clear what it means to just “look
for correlations.” Data do not speak
for themselves. So gathering lots of data will not result in eloquent models
that understand the whats that
matter. Big data sets cannot pull themselves up by the bootstraps (nothing can pull itself up by the
bootstraps!) thereby yielding useful models. So, without explicit thoughtful models
that guide the enterprise, we will be saddled with implicit models that obscure
(and trivialize) what we are doing (as noted here without good models it is
even difficult to separate good data from bad).

None of this would be worth mentioning were it not for the
mesmerizing powers of Big Data. We have seen this before (here, and here for
example). Big Data is the modern avatar
to classical empiricist methodology. It’s appeal is its promises to provide
insight without intellectual sweat. This time, however, Empiricism has found a slogan
attached to a technology, Google being the all-powerful mantra. Not
surprisingly, money-making slogans can be very enticing and Google
intellectuals (e.g. Peter Norvig) can gain powerful platforms. And though I am
quite sure that like all other (empiricist) attempts to circumvent thought,
this too will ultimately fail, it’s demise may not come soon enough to prevent
serious damage. So when you hear the siren calls of Big Data I suggest the
following prophylactic procedure; repeat Kant’s dictum to yourself, viz. data
without theory is blind, data without theory is blind, data without theory is
blind…and hope it soon goes away.

Monday, March 25, 2013

David Pesetsky gave a talk at the most recent LSA in Boston that addressed the lack of sympathy for Generative Grammar in the popular press. Indeed, it extends beyond that. Neither Science nor Nature are well know outlets of generative research. At the end of the talk, I put in my two cents. I suggested that a large obstacle to generous treatment was the fact that the most influential and prolific living linguist (Guess who? A hint: initials are NC) was also one of the leading political dissidents in the West. Much as our media love to laud dissidents over there they have little taste for those that highlight the shortcomings of our own cultural, political and financial elite. At any rate, suffice it to say that Chomsky's political writings have irritated a lot of people. These people thus have an interest in poisoning this well and one strategy is to show that even in his area of professional competence his insights have proven false and his research overthrown. This is an especially urgent task, it seems, for the liberal media like the New Yorker (see here). Glenn Greenwald has an excellent discussion of how character assassination is intended to work. Sadly, similar BS is not absent even within linguistics. I need not name names: we all know who the offenders are. So why is generative linguistics a hard sell? Well, part of the answer has nothing to do with the results of the field. What to do about it? In our small part of the world, we can refrain from character assassination and stick to debating the issues. Ideas are fair game, personalities are not.

There is a long tradition of attributing linguistic facility in humans to our greater general intelligence. This has always been a hard sell to anyone who has had a pet (a humbling experience; thanks Sampson) but it persists. The conceit has been to substitute greater general intelligence for dedicated species specific capacities. However, this tack has become harder the more we learn about animal cognition, given the increasing evidence that animals are pretty damn smart. Here is another popular review from the front. Chimps, elephants and octopi have proven themselves to be very very clever. Nonetheless, not a word from any of them (though I have been trying to get Sampson to guest blog). Smart as they are, anything remotely like our linguistic capacity is still unattested in even the smartest animals. Think that maybe being smart and being linguistically facile are unrelated?

Wednesday, March 20, 2013

Jeff Watumull send me an early version of the following and I immediately asked him if I could post it. It is a vigorous rebuttal of Postal's argument outlined in several of his more recent papers. An earlier post on a similar topic generated a lot of interest and discussion. Jeff here argues (convincingly in my view) that Platonism and the biolinguistic program are perfectly compatible. If correct, and I will let you dear readers judge for yourselves, there is even less to Postal's Platonist critique of the biolinguistic program than I earlier conceded. Jeff argues that Postal's critique is a non-sequitur even on its Platonist own assumptions. The good news: if you find Platonism appealing you can still be a good biolinguist. Whew! Thanks Jeff. Enjoy the piece. Oh yes, the post is on the long side. That's the price you pay for a comprehensive critique.

Biolinguistics
and Platonism: Contradictory or Consilient?

Jeffrey Watumull

(watumull@mit.edu)

1 Introduction

In
“The Incoherence of Chomsky’s ‘Biolinguistic’ Ontology” (Postal 2009), Postal
attacks biolinguistics as “junk linguistics” (Postal 2009: 121) with an “awful”
(Postal 2009: 114) ontology expounded in “gibberish” (Postal 2009: 118), the
“persuasive fore of [which] has been achieved only via a mixture of
intellectual and scholarly corruption” (Postal 2009: 104), whereas writings
espousing Postal’s ontology “manifest substance and quality of argument at an
incomparably higher intellectual level than [Chomsky’s]” (Postal 2009:
105). As a proponent of biolinguistics,
I am tempted to reply in kind to such invective, but to do so would be bad form
and bad science. A fallacy free and
dispassionate—if disputatious—rebuttal is necessary and proper.

For
Postal, language is a Platonic object, and therefore he concludes that the
biolinguistic assumption of a physical basis for language is “absurd”
(Postal 2009: 104). To the contrary, I
shall show Postal’s conclusion to be a non sequitur.

By
engaging in this argument, I fully expect Postal to accuse me of having “chosen
to defend something [i.e., biolinguistics] its own author [i.e., Chomsky] is
unwilling to” (Postal 2009: 105), from which two conclusions necessarily follow
in Postal’s mind: (i) I am a living
testament to Chomsky’s “intellectual and scholarly corruption” of the youth;
and (ii) “By exercising his undeniable right of silence here, Chomsky leaves
unimpeded the inference that he has not attempted a refutation because he
cannot” (Postal 2009: 105). It goes
without saying that I reject these conclusions and the premise from which they
do not follow.
(Incidentally, (i) corrupting the young has noble precedents (e.g., a
case from 399 BCE) and (ii) the argumentum a silentio is a classic fallacy.)

This
is not an apologia for Chomsky.
Biolinguistics has no single author: it is a research program pursued by
numerous individually-thinking scientists subordinate to no individual however
foundational and influential. Moreover
the theoretical and empirical contributions of the diverse subprograms in which
these scientists work are so numerous and important that none can be “dominant”
(Postal 2009: 104): in the intersection of cognitive science, linguistics, and
the formal sciences, the formal properties and functional architecture of
linguistic cognition are being specified; evolutionary biology is investigating
possible homologues/analogues of language in nonhuman animals; genetics is
discovering some of the genes entering into the development and operation of
the language faculty; neuroscience is mapping the physical substrate of
linguistic processing; and this is but a subset of the biolinguistics program
to “reinstate the concept of the biological basis of language capacities”
(Lenneberg 1967: viii).

The
subprogram I work in, call it mathematical biolinguistics, is so theoretically
and empirically eclectic that I am naturally interested in its ontology. It therefore cannot be “odd for [Postal’s]
opposite in the present exchange to be anyone other than Chomsky” (Postal 2009:
105).

In
the next section I very briefly and very informally define the biolinguistics
Postal impugns. The third section is a
rehearsal of Postal’s arguments for linguistic Platonism and ipso facto (so he assumes) against biolinguistics. I proceed in the fourth section to analyze
some of the flaws in these arguments, demonstrating that the ontologies of
Platonism and biolinguistics—properly defined—are not mutually exclusive and
contradictory, but in fact mutually reinforcing and consilient in a coherent
and compelling philosophy of language.

I
must add that my work and the ontology it assumes are not representative of all
biolinguistic research. Many would
accept my thesis that, just as engineers have encoded abstract software into
concrete hardware, evolution has encoded within the neurobiology of Homo
sapiens sapiens a formal system (computable functions) generative of an infinite
set of linguistic expressions, modulo my
understanding of the formal system as a Platonic object. Nor is mine the only coherent interpretation
of biolinguistics. So it must not be
thought that someone with my philosophy is the only possible “opposite [to
Postal] in the present exchange.”

2 Biolinguistics

Let
the ontology of some research program be defined as “biolinguistic” if it
assumes, investigates, and is informed by the biological basis of language—a
definition subsuming many productive programs of research in the formal and
natural sciences. But so general a
definition cannot adjudicate the case with Postal. At issue here is the particular definition of
biolinguistics that identifies language as I-language—i.e., a computational
system (a function in intension) internal to the
cognitive/neurobiological architecture of an individual of the species Homo
sapiens sapiens—the properties of which are determined by the three factors
that enter into the design of any biological system: genetics, external
stimuli, and laws of nature.

That
Chomsky invented the term I-language and has expatiated on
the three factors does not render him the “author” (Postal 2009: 105) of
biolinguistics—that would be a category error analogous to attributing
“authorship” of evolutionary biology to Darwin given his invention of the term natural selection and expatiation on the factors entering into
common descent with modification.
Biolinguistics and evolutionary biology are research programs to
investigate objects and processes of nature.
Thus the only author of I-language is nature. And thus anyone is free to recognize the
ontology of biolinguistics as here defined.

3 Platonist Ontology

The
incoherence of the biolinguistic ontology is claimed to derive from the fact
that “there can be no such thing” (Postal 2009: 105) as
biolinguistics, which assumes that “a mentally represented grammar and [the
language-specific genetic endowment] UG are real objects, part of the physical
world, where we understand mental states and representations to be physically
encoded in some manner [in the brain].
Statements about particular grammars or about UG are true or false
statements about steady states attained or the initial state (assumed fixed for
the species), each of which is a definite real-world object, situated in
space-time and entering into causal relations” (Chomsky 1983: 156-157). To Postal, this ontology is as “absurd”
as a “biomathematics” or a “biologic,” for “Were mathematics biological, brain
research might resolve such questions as whether Goldbach’s
Conjecture
is true. Were logic biological, one
might seek grants to study the biological basis of the validity of Modus Ponens. The ludicrous
character of such potential research is a measure of the folly of the idea that
these fields study biological things” (Postal 2009: 104, 105).

By
analogy, Postal argues that the objects of linguistic inquiry are not physical
(a fortiori not biological), but rather “like numbers, propositions, etc. are
abstract objects, hence things not located in space and time, indeed not
located anywhere.
They are also things which cannot be created or destroyed, which cannot
cause or be caused. [Natural languages]
are collections of other abstract objects normally called sentences, each of which is a set” (Postal 2009: 105).

In
the paper under consideration, Postal does not expound this ontology (see
Postal 2004); a “brief exposition of its essence” (Postal 2009: 106) suffices
for his and my purposes. Essential to
the ontology—a form of linguistic Platonism—are the type/token distinction and
discrete infinity.

3.1 Types/Tokens

“ES IST DER GEIST DER SICH DEN KORPER BAUT:
[S]uch is the nine word inscription on a Harvard museum. The count is nine because we count der
both times; we are counting concrete physical objects, nine in a row. When on the other hand statistics are
compiled regarding students’ vocabularies, a firm line is drawn at repetitions;
no cheating. Such are two contrasting
senses in which we use the word word. A word in the second sense is not a physical
object, not a dribble of ink or an incision in granite, but an abstract
object. In the second sense of the word word
it is not two words der that turn up in the inscription, but one word der
that gets inscribed twice. Words in the
first sense have come to be called tokens; words in the second sense are called
types” (Quine 1987: 216-217).

The
distinction applies to sentences: for instance, in the classic story by Dr.
Seuss, there exist (by my quick count) six tokens of the one type I do not like green eggs and ham. Postal defines sentence tokens and types as
the objects of inquiry for biolinguistics and linguistic Platonism,
respectively. For biolinguistics, as
Postal understands it, a sentence is nothing more than a “brain-internal token” (Postal 2009: 107)—a mental
representation. Such an object is
defined by spatiotemporal (neurobiological) coordinates with causes (cognitive,
chemical, etc.) and effects (e.g., in reasoning and communication). For linguistic Platonism, as Postal
understands it, this physical object is (if anything) a token of an abstract
type, with only the latter being really real. Empirically, “island constraints, conditions
on parasitic gaps, binding issues, negative polarity items, etc.” obtain not of
physical objects per se, but of abstractions: “Where is the French sentence Ça signifie quoi? — is it in France, the French Consulate in New
York, President Sarkozy’s brain? When
did it begin, when will it end? What is
it made of physically? What is its mass,
its atomic structure? Is it subject to
gravity? Such questions are nonsensical
because they advance the false presumption that sentences are physical objects”
(Postal 2009: 107). For Postal this
nonsense is nonfinite.

3.2 Discrete Infinity

“[T]he most elementary property of language—and
an unusual one in the biological world—is that it is a system of discrete
infinity consisting of hierarchically organized objects” (Chomsky 2008:
137). “Any such system is based on a
primitive operation that takes n objects already constructed,
and constructs from them a new object: in the simplest case, the set of these n
objects” (Chomsky 2005: 11). Call [the operation]
Merge.
Operating without bounds, Merge yields a discrete infinity of structured
expressions” (Chomsky 2007: 5).

Postal
invokes the type/token distinction in his critique of this biolinguistic
conception of discrete infinity. He
assumes that any object constructed by a physical system must be physical:
“Consider a liver and its production of bile, a heart and its production of
pulses of blood; all physical and obviously finite. And so it must be with any cerebral physical
production” (Postal 2009: 109). Thus if
language is a physical (neurobiological) system, then its productions
(sentences) must be physical (neurobiological tokens). But physical objects are by definition
bounded by the finiteness of spatiotemporal and operational resources: “There
is for Chomsky thus no coherent interpretation of the collection of brain-based
expressions being infinite, since each would take time and energy to construct,
[...] store, process, or whatever[...]; they have to be some kind of tokens”
(Postal 2009: 109, 111). More
abstractly, a discretely (denumerably) infinite set is one with expressions
(members) that can be related one-to-one with the expressions of one of its
subsets (and with the natural numbers).
But if language is a neurobiological system, hence finite, then
obviously it cannot contain or construct a set that can be related to the
(countable) infinity of natural numbers: “every physical production takes time,
energy, etc. and an infinite number of them requires that the physical universe
be infinite and, internal to Chomsky’s assumptions, that the brain be” (Postal
2009: 111). Reductio ad absurdum,
supposedly.

If
biolinguistics implies that expressions are bounded by the spatiotemporal and
operational resources of neurobiology, then the (infinite) majority of
expressions contained in the discrete infinity are generable only in principle:
there exist infinitely many more possible sentences than can ever be generated
in the physical universe. So for the
biolinguistic system to be defined as discretely infinite, it must be defined
as an idealization: a system abstracted away from the contingent nature of the
spatiotemporal and operational resources of neurobiology. In other words, the biolinguistic system is
discretely infinite only if abstracted from biology. And this, Postal concludes, is the
fundamental fallacy:

If “the biological [Merge function] ‘ideally’
generates an infinite collection, most of the ‘expressions’ in the collection
cannot be physical objects, not even ones in some future, and the [natural
language] cannot be one either. [A]lmost
all sentences are too complex and too numerous [...] to be given a physical
interpretation[...]. In effect, a
distinction is made between real sentences and merely ‘possible’ ones, although
this ‘possibility’ is unactualizable ever in the physical
universe. According to the biological
view, [...] the supposedly ‘possible’ sentences are, absurdly, actually
biologically impossible. Thus internal
to this ‘defense’ of Chomsky’s biolinguistic view, the overwhelming majority of
sentences cannot be assigned any reality whatever
internal to the supposed governing ontology. This
means the ontology can only claim [natural language] is infinite because,
incoherently, it is counting things the ontology cannot recognize as real”
(Postal 2009: 111).

If,
however, tokens as physical objects can implement abstract types, then
presumably a recursive rule—a finite type—could be tokenized as a procedure in
the mind/brain. This Postal concedes:
although “nothing physical is a rule or recursive,” because recursive rules are
Platonic, a “physical structure can encode rules” (Postal 2009: 110). Presumably therefore Merge—the
mentally-represented/neurobiologically-implemented recursive procedure posited
in biolinguistics to generate discrete infinity—is a legitimate posit. But Postal objects: “an interpretation of
physical things as representing particular abstractions [is] something
Chomsky’s explicit brain ontology has no place for” (Postal 2009: 110). Furthermore, Merge is supposed to generate
sets, and sets are Platonic abstractions, but as “an aspect of the
spatiotemporal world, [Merge] cannot ‘generate’ an abstract object like a set”
(Postal 2009: 114). So Merge is either
biological—not mathematical and hence incapable of generating a set (let alone
an infinite one)—or it is mathematical—hence nonbiological but capable of
generating discrete infinity. In sum,
language is either physical or it is Platonic, and
only under the latter definition can it be predicated of that “most elementary
property,” discrete infinity—or so Postal maintains.

4 Mathematical Biolinguistic Ontology

Let
me affirm at the outset my commitment to mathematical Platonism, which
informs my biolinguistic ontology in ways to be discussed. More strongly than Chomsky, who does grant
mathematical Platonism “a certain initial plausibility,” I am convinced of the
existence of “a Platonic heaven [of] arithmetic and [...] set theory,” inter
alia, that “the truths of arithmetic are what they are, independent of any
facts of individual psychology, and we seem to discover these truths somewhat
in the way that we discover facts about the physical world” (Chomsky 1986:
33). It follows from this position that
I must be committed to linguistic Platonism for any
linguistic objects reducible to or properly characterized as mathematical
objects. And indeed in my theory of
natural language (see Watumull 2012), the quiddities that define a system as
linguistic are ultimately mathematical in nature. (The “essence” of language, if you will, is
mathematical—a proposition I shall not defend here, assuming it to be
essentially correct, for at issue in this discussion is not whether the
proposition is true, but whether it is consistent with a biolinguistic ontology
if true.)

4.1 Overlapping Magisteria

I
and others (see, e.g., Hauser, Chomsky, Fitch 2002; Watumull, Hauser, Berwick
2013) posit a recursive function generative of structured sets of expressions
as central to natural language; this function is defined in intension as
internal to the mind/brain of an individual of the species Homo sapiens
sapiens. So conceived, I-language has
mathematical and biological aspects.

Nonsense!
Postal would spout: The ontologies
of mathematics and biology are nonoverlapping magisteria! Assuming mathematical Platonism, I concur
that a mathematical object per se such as a recursive function (the type) is
not physical. However even Postal (2009:
110) concedes that such an object can be physically encoded (as a token). The rules of arithmetic for instance are multiply realizable, from the analog abacus to the digital computer
to the brain; mutatis mutandis for other functions, sets, etc. And mutatis mutandis for abstract objects
definable as mathematical at the proper level of analysis, such as a computer
program:

“You know that if your computer beats
you at chess, it is really the program that has beaten you,
not the silicon atoms or the computer as
such. The abstract program is
instantiated physically as a high-level behaviour
of vast numbers of atoms, but the explanation of why it has beaten
you cannot be expressed without
also referring to the program in its own right.
That program has also been instantiated, unchanged, in a long chain of different physical
substrates, including neurons in the brains of the programmers and radio waves when you downloaded the
program via wireless networking, and finally as states of long- and short-term memory banks in your computer. The specifics of that chain of instantiations
may be relevant to explaining how the
program reached you, but it is irrelevant to why it beat you: there, the
content of the knowledge (in it, and
in you) is the whole story. That story
is an explanation that refers ineluctably to abstractions;
and therefore those abstractions exist, and really do affect physical objects
in the way required by the explanation”
(Deutsch 2011: 114-115).

(Though
I shall not rehearse the argument here, I am convinced by Gold (2006) that
“mathematical objects may be abstract, but they’re NOT [necessarily] acausal”
because they can be essential to—ineliminable from—causal explanations. The potential implications of this thesis for
linguistic Platonism are not uninteresting.)

I
take the multiple realizability of the chess program to evidence the reality of
abstractions as well as anything can (and I assume Postal would agree):
something “substrate neutral” (Dennett 1995) is held constant across multiple
media. That something I submit is a
computable function; equivalently, that constant is a form of Turing machine
(the mathematical abstraction representing the formal properties and functions
definitional of—and hence universal to—any computational system).

4.2 The Linguistic Turing Machine

Within
mathematical biolinguistics, it has been argued that I-language is a form of
Turing machine (see Watumull 2012; Watumull, Hauser, Berwick 2012), even by
those Postal diagnoses as allergic to such abstractions:

“[E]ven though we have a finite brain, that
brain is really more like the control unit for an infinite computer. That is, a finite automaton is limited
strictly to its own memory capacity, and we are not. We are like a Turing machine in the sense
that although we have a finite control unit for a brain, nevertheless we can
use indefinite amounts of memory that are given to us externally[, say on a
“tape,”] to perform more and more complex computations[...]. We do not have to learn anything new to
extend our capacities in this way” (Chomsky 2004: 41-42).

As
Postal would observe, this “involves an interpretation of physical things as
representing particular abstractions,” which he concedes is
coherent in general because obviously “physical structure can encode rules” and
other abstract objects (e.g., recursive functions) (Postal 2009: 110)—computer
programs, I should say, are a case in point.

4.3 Idealization

Postal
(2012: 18) has dismissed discussion of a linguistic Turing machine as
“confus[ing] an ideal machine[...], an abstract object, with a machine, the
human brain, every aspect of which is physical.” I-language qua Turing machine is obviously an
idealization, with its unbounded running time and access to unbounded memory,
enabling unbounded computation. And
obviously “[unboundedness] denotes something physically counterfactual as far
as brains and computers are concerned.
Similarly, the claim ‘we can go on indefinitely’ [...] is subordinated
to the counterfactual ‘if we just have more and more time.’ Alas we do not, so we can’t go on
indefinitely” (Postal 2012: 18). Alas it
is Postal who is confused.

4.3.1 Indefinite Computation

Postal’s
first confusion is particular to the idealization of indefinite
computation. Consider arithmetic. My brain (and presumably Postal’s) and my
computer encode a program (call it ADD) that determines functions of the form fADD(X + Y) = Z (but not W)
over an infinite range. Analogously, my
brain (and Postal’s) but not (yet) my computer encodes a program (call it
MERGE) that determines functions of the form fMERGE(α, β) = {α, β}—with
syntactic structures assigned definite semantic and phonological forms—over an
infinite range. These programs are of
course limited in performance by spatiotemporal constraints, but the programs
themselves—the functions in intension—retain their deterministic form even as
physical resources vary (e.g., ADD determines that 2 + 2 = 4 independent of
performance resources).

Assuming
a mathematical biolinguistic ontology, I-language is a
cognitive-neurobiological token of an abstract type; it “generates” sets in the
way axioms “generate” theorems. As the
mathematician Gregory Chaitin observes, “theorems are compressed into the axioms” so that “I think of axioms as a computer
program for generating all theorems” (Chaitin 2005: 65). Consider how a computer program explicitly
representing the Euclidean axioms encodes only a finite number of bits; it does
not—indeed cannot—encode the infinite number of bits that could be derived from
the postulates, but it would be obtuse to deny that such an infinity is
implicit (compressed) in the explicit axioms.
Likewise, zn+1 = zn2 + c defines the Mandelbrot
set (as I-language defines the set of linguistic expressions) so that the
infinite complexity of the latter really is implicitly represented in the
finite simplicity of the former.

So
while it is true that physically we cannot perform indefinite computation, we
are endowed physically with a competence that does define a set that could
be generated by indefinite computation.
(A subtle spin on the notion of competence perhaps more palatable to
Postal defines it as “the ability to handle arbitrary new cases when they
arise” such that “infinite knowledge” defines an “open-ended response
capability” (Tabor 2009: 162).) Postal
must concede the mathematical truth that linguistic competence, formalized as a
function in intension, does indeed define an infinite set. However, he could contest my could
as introducing a hypothetical that guts biolinguistics of any biological
substance, but that would be unwise.

Language
is a complex phenomenon: we can investigate its computational (mathematical)
properties independent of its biological aspects just as legitimately as we can
investigate its biological properties independent of its social aspects (with
no pretense to be carving language at its ontological joints). In each domain, laws—or, at minimum, robust
generalizations—license counterfactuals (as is well understood in the
philosophy of science). In discussing
indefinite computation, counterfactuals are licensed by the laws expounded in
computability theory:

“[T]he question whether a function is
effectively computable hinges solely on the behavior of that function in
neighborhoods of infinity[...]. The
class of effectively computable functions is obtained in the ideal case where
all of the practical restrictions on running time and memory space are
removed. Thus the class is a theoretical
upper bound on what can ever in any century be considered computable” (Enderton
1977: 530).

A
theory of linguistic competence establishes an “upper bound,” or rather
delineates the boundary conditions, on what can ever be considered a linguistic
pattern (e.g., a grammatical sentence).
Some of those patterns extend into “neighborhoods of infinity” by the
iteration of a recursive function.
Tautologically, those neighborhoods are physically inaccessible, but
that is irrelevant. What is important is
the mathematical induction from finite
to infinite:
Merge applies to any two arguments to form a a set containing those two
elements such that its application can only be bounded by stipulation. In fact a recursive function such as Merge
characterizes the “iterative conception of a set,” with sets of discrete
objects “recursively generated at each stage,” such that “the way sets are
inductively generated” is formally equivalent to “the way the natural numbers
[...] are inductively generated” (Boolos 1971: 223).

The
natural numbers are subsumed in the computable numbers, “the real numbers whose
expressions as a decimal are calculable by finite means” (Turing 1936:
230). (The phrase “finite means” should
strike a chord with many language scientists.)
It was by defining the computable
numbers that Turing proved the coherency of a finitary procedure generative of
an infinite set.

“For instance, there would be a machine to
calculate the decimal expansion of π[...].
π being an infinite decimal, the work of the machine would never end,
and it would need an unlimited amount of working space on its ‘tape’. But it would arrive at every decimal place in
some finite time, having used only a finite quantity
of tape. And everything about the
process could be defined by a finite table[...]. This meant that [Turing] had a way of
representing a number like π, an infinite decimal, by a finite
table. The same would be true of the
square root of three, or the logarithm of seven—or any other number defined by
some rule” (Hodges 1983: 100).

Though
they have not been sufficiently explicitly acknowledged as such, Turing’s
concepts are foundational to the biolinguistic program. I-language is a way of representing an infinite
set by a finite table (a function). The
set of linguistic expressions being infinite, “the work of the machine would
never end,” but Postal must concede that nevertheless I-language “would arrive
at every [sentence] in some finite time, having used only a finite quantity of
tape. And everything about the process
could be defined by a finite table.”
This gives a rigorous sense to the linguistic notion “infinite use of finite
means.”

4.3.1.1 Generation and Explanation

But
for all the foregoing, the finitude/infinitude distinction is not so
fundamental given the fact that “[a] formal system can simply be defined to be
any mechanical procedure for producing formulas” (Gödel 1934: 370). The infinitude of the set of expressions
generated is not as fundamental as the finitude of I-language (the generative
function) for the following reason: it is only because the function is finite
that it can enumerate the elements of the set (infinite or not); and such a
compact function could be—and ex hypothesi is—neurobiologicall encoded. Even assuming Postal’s ontology in which
“[natural languages] are collections of [...] abstract objects” (Postal 2009:
105), membership in these collections is granted (and thereby constrained) by
the finitary procedure, for not just any (abstract) object qualifies. In order for an object to be classified as
linguistic, it must be generated by I-language; in other words, to be a
linguistic object is to be generated by I-language. And thus I-language explains why a given natural language contains the member
expressions it does.

This
notion of I-language as explanation generalizes to the
notion of formal system as scientific theory:

“I think of a scientific theory as a binary
computer program for calculating observations, which are also written in
binary. And you have a law of nature if
there is compression, if the experimental data is compressed into a computer
program that has a smaller number of bits than are in the data that it
explains. The greater the degree of compression,
the better the law, the more you understand the data. But if the experimental data cannot be
compressed, if the smallest program for calculating it is just as large as it
is [...], then the data is lawless, unstructured, patternless, not amenable to
scientific study, incomprehensible. In a word, random, irreducible” (Chaitin
2005: 64).

This
notion is particularly important, as Turing (1954: 592) observed, “[w]hen the
number is infinite, or in some way not yet completed [...],” as it is for the
discrete infinity (unboundedness) of language; “a list of answers will not
suffice. Some kind of rule or systematic
procedure must be given.” Otherwise the
list is arbitrary and unconstrained. So
for linguistics, in reply to the question “Why does the infinite natural
language L contain the expressions it does?” we answer
“Because it is generated by the finite I-language f.” Thus I-language can be conceived of as the
theory explicative of linguistic data because it is the mechanism (Turing
machine) generative thereof.

4.3.2 “the thing in itself”

Second,
with respect to idealization generally, for mathematical biolinguistics to have
defined I-language as a Turing machine is not to have confused the physical
with the abstract, but rather to have abstracted away from the contingencies of
the physical, and thereby discovered the mathematical constants that must of
necessity be implemented for any system—here biological—to be linguistic (on my
theory). This abstraction from the
physical is part and parcel of the methodology and, more importantly, the metaphysics of normal science, which proceeds by the
“making of abstract mathematical models of the universe to which at least the
physicists give a higher degree of reality than they accord the ordinary world
of sensation” (Weinberg 1976: 28). The
idealization is the way things really are. Consider Euclidean objects: e.g.,
dimensionless points, breadthless lines, perfect circles, and the like. These objects do not exist in the physical
world. The points, lines, and circles
drawn by geometers are but imperfect approximations of abstract Forms—the
objects in themselves—which constitute the
ontology of geometry. For instance, the
theorem that a tangent to a circle intersects the circle at a single point is
true only of the idealized objects; in any concrete representation, the
intersection of the line with the circle cannot be a point in the technical
sense as “that which has no part,” for there will always be some overlap. As Plato understood (Republic VI: 510d), physical reality is an intransparent and
inconstant surface deep beneath which exist the pellucid and perfect constants
of reality, formal in nature:

“[A]lthough
[geometers] use visible figures and make claims about them, their thought isn’t
directed to them but to the originals of which these figures are images. They make their claims for the sake of the Square
itself and the Diagonal itself, not the particular square and diagonal they
draw; and so on in all cases. These
figures that they make and draw, of which shadows and reflections in water are
images, they now in turn use as images, in seeking to behold those
realities—the things in themselves—that one cannot comprehend except by means
of thought.”

Analogously,
any particular I-language (implemented in a particular mind/brain) is an
imperfect representation of a form (or Form) of Turing machine. But, Postal would object,
the linguistic Turing machine is Platonic, hence nonbiological, and hence bio-linguistics
is contradictory. But,
I should rebut, this objection is a non sequitur.

I am
assuming (too strongly perhaps) that fundamentally a system is linguistic in
virtue of mathematical (nonbiological) aspects.
Nevertheless, in our universe, the only implementations of these mathematical
aspects yet discovered (or devised) are biological; indeed the existence of
these mathematical systems is known to us only by their biological
manifestations—i.e., in our linguistic brains and
behaviors—which
is reason enough to pursue bio-linguistics. To borrow some rhetorical equipment, biology
is the ladder we climb to the “Platonic heaven” of linguistic Forms, though it
would be scientific suicide to throw the ladder away once up it. That chance and necessity—biological
evolution and mathematical Form—have converged to form I-language is an
astonishing fact in need of scientific explanation. It is a fact that one biological system
(i.e., the human brain) has encoded within it and/or has access to Platonic
objects. (Postal must assume that our
finite brains can access an infinite set of Platonic sentences. The ontological status of the latter is not
obvious to me, but obviously I am committed to the existence of the encoding
within our brains of a finite Platonic function for unbounded computation.) Surely a research program formulated to
investigate this encoding/access is not perforce incoherent.

I do
however deny any implication here that such complex cognition, “in some most
mysterious manner, springs only from the organic
chemistry, or perhaps the quantum mechanics, of processes that take place in
carbon-based biological brains. [I] have
no patience with this parochial, bio-chauvinistic view[:] the key is not the stuff
out of which brains are made, but the patterns that can come to exist
inside the stuff of a brain” (Hofstadter 1999 [1979]: P-4, P-3). Thus, as with chess patterns, it is not by
necessity that linguistic patterns spring from the stuff of the brain; but the
fact remains that they can and do. And
thus linguistics is just as much a biological science as it is a formal
science.

To
reiterate, at present there exists no procedure other than human intuition to
decide the set of linguistic patterns.
The neurobiology cannot answer the question whether some pattern is
linguistic (e.g., whether some sentence is grammatical), but it encodes the
procedure that enables the human to intuit the answer to such a question. Analogously, neurobiological research would
not establish the truth of Goldbach’s conjecture or the validity of reasoning
by modus ponens, but rather would be unified with research in cognitive science
to establish (discover) the rules and representations encoded neurobiologically
that enable cognitive conjecture and reasoning.

4.4 Encoding Abstract Objects in Physical
Systems

Postal
believes that an “explicit brain ontology” as assumed in biolinguistics “has no
place for” the encoding of an abstract object such as a Turing machine in a
physical system such as the brain—but I see no
grounds whatsoever
for this belief. Not only is this belief
contradicted by Chomsky’s Turing machine analogy, but Postal himself quotes
Chomsky discussing how in biolinguistics “we understand mental states and
representations to be physically encoded in some manner” (1983: 156-157); and
to physically encode something presumes a non-physical something to be so
encoded. For this reason “it is the
mentalistic studies that will ultimately be of greatest value for the
investigation of neurophysiological mechanisms, since they alone are concerned
with determining abstractly the properties that such mechanisms must exhibit
and the functions they must perform” (Chomsky 1965: 193).

It is in
this sense of neurobiology encoding mathematical properties and functions that,
“astonishingly” (Postal 2012: 23), we observe the obvious fact that “We don’t
have sets in our heads. So you have to
know that when we develop a theory about our thinking, about our computation,
internal processing and so on in terms of sets, that it’s going to have to be
translated into some terms that are neurologically realizable. [Y]ou talk about a generative grammar as
being based on an operation of Merge that forms sets, and so on and so
forth. That’s something metaphorical,
and the metaphor has to be spelled out someday” (Chomsky 2012: 91). In other words, while the formal aspects of a
Turing machine (e.g., Merge, sets, etc.) are, ex hypothesi, realized
neurologically, it would be absurd (astonishing) to expect physical
representation of our arbitrary notations (e.g., fMERGE(X, Y) = {X, Y}). As
Turing observed, in researching the similarities of minds and machines, “we
should look [...] for mathematical analogies of function” (1950:
439)—similarities in software, not hardware.

Of
course an ontological commitment to abstract properties and functions is not
necessarily a commitment to Platonism (as Aristotle demonstrated and many in
the biolinguistics program would argue), but it is certainly the default
setting. So it can be argued that
I-language is just like Deutsch’s chess program: a multiply realizable
computable function (or system of computable functions). Indeed given that a Turing machine is a
mathematical abstraction, I-language qua
Turing machine is necessarily and properly defined as a physically
(neurobiologically) encoded Platonic object.

5 Conclusion

I
have argued that mathematical biolinguistics is based on the perfectly coherent concept of computation—as formulated by
Turing—unifying mathematical Platonism and biolinguistics: evolution has
encoded within the neurobiology of Homo sapiens sapiens a formal system
(computable function(s)) generative of an infinite set of linguistic
expressions (just as engineers have encoded within the hardware of computers
finite functions generative of infinite output). This thesis, I submit, is or would be
accepted by the majority of researchers in biolinguistics, perhaps modulo the
Platonism, for indeed it is not necessary to accept the reality of mathematical
objects to accept the reality of physical computation. However, I am a mathematical Platonist, and
thus do recognize the reality of mathematical objects, and thus do argue
I-language to be a concretization (an “embodiment” in the technical sense) of a
mathematical abstraction (a Turing machine), which to my mind best explains the
design of language.

References

Boolos,
George S. 1971. The iterative conception of set. The
Journal of Philosophy
68: 215-231.